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NASA/CRm2001-210810 2001--GT-0165
Two-Equation Turbulence Models forPrediction of Heat Transfer on a
Transonic Turbine Blade
Vijay K. Garg and Ali A. Ameri
AYT Corporation, Brook Park, Ohio
April 2001
https://ntrs.nasa.gov/search.jsp?R=20010047410 2020-05-10T03:39:06+00:00Z
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NASA/CRm2001-210810 2001-GT-0165
Two-Equation Turbulence Models forPrediction of Heat Transfer on a
Transonic Turbine Blade
Vijay K. Garg and Ali A. Ameri
AYT Corporation, Brook Park, Ohio
Prepared for the
2001 Turbo Expo
sponsored by the American Society of Mechanical Engineers
New Orleans, Louisiana, June 4-7, 2001
Prepared under Contract NAS3--00180
National Aeronautics and
Space Administration
Glenn Research Center
April 2001
Acknowledgments
The authors wish to thank Dr. Raymond Gaugler, Chief, Turbine Branch at the NASA Glenn Research Center
for his support of this work.
NASA Center for Aerospace Information7121 Standard Drive
Hanover, MD 21076Price Code: A03
Available from
National Technical Information Service
5285 Port Royal Road
Springfield, VA 22100Price Code: A03
Available electronically at http: //g!trs.grc.nasa.gov/GLTRS
TWO-EQUATION TURBULENCE MODELS FOR PREDICTION OFHEAT TRANSFER ON A TRANSONIC TURBINE BLADE
Vijay K. Garg and Ali A. AmerlAYTCorporation
2001 Aerospace ParkwayBrook Park, Ohio 44142
ABSTRACT
Two versions of the two-equation k-o) model and a shear stress
transport (SST) model are used in a three-dimensional, multi-block,
Navier-Stokes code to compare the detailed heat transfer measurements
on a transonic turbine blade. It is found that the SST model resolves
the passage vortex better on the suction side of the blade, thus yielding
a better comparison with the experimental data than either of the k-c0
models. However, the comparison is still deficient on the suction side
of the blade. Use of the SST model does require the computation of
distance from a wall, which for a multi-block grid, such as in the
present case, can be complicated. However, a relatively easy f'Lx for
this problem was devised. Also addressed are issues such as (1)
computation of the production term in the turbulence equations for
aerodynamic applications, and (2) the relation between the
computational and experimental values for the turbulence length scale,
and its influence on the passage vortex on the suction side of the
turbine blade.
NOMENCLATURE
Cp
Fj, F 2k
M
Pr
r
Re
Ril
s
S
St
T
Tu
U
U
specific heat at constant pressure
constants defined in Eqs. (7) and (10)
turbulence kinetic energy; also thermal conductivity in Eq. (20)
turbulence length scale
Mach number
Prandtl number
recovery factor = Pr T_3(for turbulent flow)
Reynolds number based on the blade axial chord and U_,
auto-correlation tensor for velocity
distance from the leading edge along the pressure or suction
surface
strain rate
Stanton number defined in Eq. (20)
temperature
turbulence intensity
velocity
magnitude of velocity
V*
x,y,z
y+
I_,lrAyE
,/
1(
V
0G
"C
CO
shear velocity
Cartesian coordinate system with origin at the geometric
stagnation point, and z in the spanwise direction
distance in wall coordinates (= yv'/v)
thermal diffusivity
constant ha Eqs. (8) and (9)
constants given in Eqs. (8) and (12)
distance (from the wall) of the first point off the wall
turbulence dissipation rate
constant given in Eqs. (8) and (12); also ratio of specific heats
in Eq. (21)
von Karman constant given in Eq. (8)
viscosity
kinematic viscosity
density
constants given in Eqs. (8) and (12)
shear stress
specific turbulence dissipation rate (= e/k)
absolute value of vorticity
Subscriptsaw adiabatic Wall value
ef effective value
ex value at exit
exp experimental valuein value at inlet
is isentropic valuelaminar value
o stagnation valuet turbulent value
w value at wall
INTRODUCTION
Accurate prediction of turbine blade heat transfer, so crucial to the
efficient design of blade cooling schemes, still remains a challenging
task despite a lot of work in this area. The main cause for the lack of
NASA/CR--2001-210810 1
agreement with experimental data in such predictions is usually cited
to be the turbulence modeling. This is due to a variety of flow and
heat transfer phenomena which are encountered in turbine passages.
Stagnation flow heat transfer, heat transfer in the presence of steep
pressure gradients both favorable and adverse, free stream turbulence,
high Mach number and three-dimensional effects are only some of the
items in a long list of phenomena present in these passages.
Well documented data sets exist which test the capabilities of
numerical schemes for the prediction of blade heat transfer. The
NASA Glenn transonic blade (Giel et al., 1999) is an example of such
a challenging test case that will be examined in this work. It is a data
set for a rotor cascade with a large turning angle. The experimental
heat transfer measurements are available for the blade at three
spanwise locations for varying exit Mach number, Reynolds number,
inlet turbulence intensity as well as the inlet boundary layer thickness
to the cascade. Some of the cases are highly three-dimensional and
pose a severe challenge to the predictive ability of any numericalscheme.
As the direct numerical simulation of such flows are not
anticipated to become routine for many more years, turbulence
modeling seems to remain the only option. By far the most popular
turbulence models utilized today for flow and heat transfer calculations
are the low-Reynolds number two-equation eddy viscosity models.
The k-e and k-to are the most utilized models. These models often
offer a good balance between complexity and accuracy. The ability to
mimic transition to turbulence which often is present on turbine blades
and the ability to integrate to the walls are other reasons for their
popularity. These models have been applied to a variety of
experimentally measured cases and their accuracy assessed, yet they do
not offer good comparisons consistently. There are also questions
regarding the modeling of the stagnation heat transfer and the free-
stream turbulence, and the best manner to specify them as boundary
conditions.
It was found during the course of this work that the low-Reynolds
number version of the k-to turbulence model adopted as the basic
model in the Glenn-HT code was inadequate in predicting the blade
surface heat transfer for this case. Thus, use of other eddy viscosity
models was explored. Our preference for the k-to model stems from
its robustness and absence of the distance to the wall in its formulation,
which makes it attractive for a multi-block code. The models tested
other than the original k-to model (Wilcox, 1988) were the 1998 k-co
model (Wilcox, 1998) and Menter's SST model (1994). Menter's SST
model among these performed the best. This model was devised by
combining good near-wall behavior of the Wilcox's k-to model, and the
k-e model (Jones and Launder, 1973) away from the walls. Menter
reports much improved agreement with experiments (Menter, 1994,
1996) for velocity profiles, pressure distributions and skin friction
distributions for a variety of test cases. The SST model does require
the computation of distance to the wall in its formulation, the
implementation of which is discussed later in the paper.
The computation of the leading edge heat transfer also requires
special attention. It has been reported by Menter (1994) and others
that the use of eddy viscosity models causes overprediction of the
turbulence production term. This in turn leads to overprediction of the
leading edge heat transfer. Two "fixes" for this have been suggested
by Menter (1992) and by Kato and Launder (1993). These have been
tested in this work.
The experiment under consideration provides both the inlet
turbulence intensity and length scale in the form of an integral length
scale. As many experiments do not provide the length scale, the CFD
practitioner feels free to choose a value, often one that best fits the
experimental data (Moore and Moore, 1999). Moss and Oldfield
(1992) studied the effect of free-stream turbulence scale on heat
transfer over a flat plate and concluded that the heat transfer coefficient
cannot be predicted accurately using turbulence level alone, but a
reasonable prediction can be made using the turbulence level and the
integral scale even for highly anisotropic turbulence at high intensities.
Van Fossen et al. (1995) also found that the turbulence length scale
affects the stagnation region heat transfer. Even when the
experimental length scale is known, such as in the present case, the
computational value may not be the same. This is described later on.
For flow over a blade, the turbulence length scale also affects the
passage vortex on the suction side of the blade - an essentially three-
dimensional phenomenon. It is therefore essential to use a correct
value for the turbulence length scale.
In this paper following the introduction, equations for the
turbulence models considered are presented. Next the computation of
production term in the turbulence model equations is discussed, and the
relation between the experimental and computational value of the
turbulence length scale is defined. Their effects on sample calculations
are then examined. Subsequently the results of the calculations with
the various models are presented and compared with the experimental
data.
ANALYSIS
The numerical simulation has been performed using the NASA
Glenn Research Center General Multi-Block Navier-Stokes Convective
Heat Transfer code, Glenn-HT. Briefly, the code, formerly known as
TRAF3D.MB (Steinthorsson et al., 1997), is an explicit, multigrid, cell-
centered, finite volume code with a k-to turbulence model without any
wall functions. This is a general purpose flow solver designed for
simulations of flows in complicated geometries. The Navier-Stokes
equations in a rotating Cartesian coordinate system are mapped onto
a general body-fitted coordinate system using standard techniques. The
multistage Runge-Kutta scheme developed by Jameson et al. (1981) is
used to advance the flow solution in time from an initial approximation
to the steady state. A spatially varying time step along with a CFL
number of 4 is used to speed convergence to the steady state.
Eigenvalue-scaled artificial dissipation and variable-coefficient implicit
residual smoothing are used along with a full-multigrid method. The
overall accuracy of the code is second order. No wall functions are
used, thus avoiding any bias to the complex three-dimensional flow
structures near the blade or any other surface.
While the Glenn-t-IT code has the original k-to model (Wilcox,
1988), the shear stress transport (S ST) model of Menter (1994), and the
k-to model of Wilcox (1998) were implemented in it for comparing the
experimental heat transfer data of Giel et al. (1999) on a transonic
turbine rotor. The SST model encompasses both the k-to and the k-e
models, with the original k-to model of Wilcox (1988) activated in the
near-wall region and the standard k-e model (Jones and Launder, 1973)
activated in the outer wake region and in free shear layers. Moreover,
NASA/CR--2001-210810 2
thedefinitionofeddyviscosityismodifiedto account for the transport
of the principal turbulent shear stress. The reader is referred to Menter
(1994) for an elucidating discussion of the SST model. Following
Menter (1994), the equations for the SST model can be written as
Dt - xq a--_/ - 13"p°k _ _ ÷ °k_r)(1)
D(po) =__y _ ___13po2+__ (_+a,_ta,)Dt v t Oxj Oxj
1 akato
+2p(l - Ft)o,_ 2 _o Ox/ Ox1
(2)
where the shear stress is given by
( 0% + Out 20uk ] 2
• °kS"(3)
8,j being the Kronecker delta. Note that the production term in Eqs.
(1) or (2) can be written as
(4)
where the strain rate tensor is given by
So = Ou_.2'+ ___2axj &,
(5)
It is known that in aerodynamic applications, use of S2 in Eq. (4) leads
to very high heat transfer coefficient at the leading edge of a blade.
To avoid this, use of f_2 or SO is recommended (Menter 1992; Kato
and Launder 1993), where f_ is the absolute value of the vorticity.
These suggestions follow from the fact that for a stagnation flow, 92 =
0 and for a simple shear flow, use of $92 is identical to that of 3a.
Based upon comparison with experimental data, we recommend the use922of $92 and will later describe the effect of using S", $92 or in Eq.
(4) on the heat transfer coefficient at the blade surface.
If _, represents any constant in the original k-co model (6k_ .... ),
and _2 anY constant in the transformed k-e model (Ok2.... ), then _, the
corresponding constant of the new model given by Eqs. (1) and (2) is
: FI ¢I + (I - FI)_2 (6)
where
_, =tanh(a,-g_)
O.09coy ' y2co CDk_,y2J
( 1 Oka¢° 10-2°)CDk, o =max 2Ors o dxj ax i
(7)
y is the distance to the next surface, and CDkt o is the positive portion
of the cross-diffusion term of Eq. (2). The choice of terms within arg;
is detailed in Menter (1994). As arg; goes to zero near the boundary-
layer edge, so does F; so that the standard k-e model is used in that
region. For the SST model, the various constants are:
oh =0.85, o_ =0.5, _x =0.075, a I=0.31
13" = 0.09, r :0.41, Yl = _iI13"- e,o _/'f_:
cr_= 1.0, o,o_: 0.856, 132: 0.0828
(8)
and the eddy viscosity is defined as
a_kv, = (9)
max(a_ _, k')F2)
where F_ is given by
: tanh(arg_) arg 2 = max(2 x/k 500____v]F_' _ 0.09toy' ),zoo )
(10)
We may note that Eqs. (7) and (I0) require the computation of y,
the distance to a wall. This can be complicated for a multi-block grid,
such as in the present case. A simple remedy, however, is to set lly
to zero for all grid cells initially, and to compute l/y once only for
those blocks that have a wall boundary condition. In fact, if a block
has more than one wall, one can specify
I = 1 + ....1 + (II)
Y Y_ Yz
where y_ is the distance of a grid cell from one wall, y_ is the distance
of the same cell from the second wall, and so on. It is easy to use any
other combination in Eq. (11). However, a negligible difference in the
heat transfer coefficient on the blade surface was found whether Eq.
(11) was used or lly was taken to be the maximum of lly_ and l/y z for
a block limited by the blade and the hub. This scheme works since for
the SST model, the k-o model is activated in the near-wall region
while the standard k-e model is activated in the outer wake region and
in free shear layers. We realize that this scheme may have some
limitations based on how various blocks are configured in the grid.
With F_ : 1, Eqs. (1) and (2) yield the k-o model where,
following Wilcox (1988), the constants are
o_,=0.5, o_, =0.5, 13_=0.075
l_*= 0.09, V_ : 519, v, -'-k/co
(12)
for Wilcox's 1988 model. For Wilcox's 1998 model, I_, = 0.072, y_ =
0.52, and there are a few more modifications (cf. Wilcox, 1998, p.
121). It may be noted that the low-Re version of the Wilcox's k-o
models was used, for which the constants ]3' and y, are modified
(Wilcox, 1998, p. 198).
It is assumed that the effective viscosity for turbulent flows can
be written as
NASA/CR--2001-210810 3
_t,f--p.,+ l_t(13)
where the laminar viscosity p_ is calculated using a power-law for its
dependence on temperature (Schlichting, 1979). The turbulent
viscosity _ is computed using the SST or the low-Re k-co model
described above. The turbulent thermal diffusivity is computed from
_, (14)a t =
p Pr_
where a constant value of 0.9 is used for the turbulent Prandtl number,
Pr t .
Boundar)' ConditionsAt the main flow inlet boundary located at an axial distance equal
to the blade axial chord upstream of the blade leading edge, the total
temperature, total pressure, whirl, and meridional flow angle are
specified, and the upstream-running Riemann invariant based on the
total absolute velocity is calculated at the first interior point and
extrapolated to the inlet. The velocity components are then decoupled
algebraically, and the density is found from total temperature, total
pressure and total velocity using an isentropic relation. For the
turbulence model, the value of k and co is specified using the
experimental conditions, namely
k = 1.5(U., TUin)2 , to = klnl_ , (15)
where Tu_. is the intensity of turbulence at the inlet (taken to be 0.09
or 0.0025 as per experimental data for the rotor), Um is the absolute
velocity at inlet, and Q is the integral length scale representing the size
of the energy containing eddies. This length scale is usually different
from that reported as part of the experimental conditions, and needs to
be revised as detailed later.
At the main flow exit plane located at an axial distance equal to
80% of the blade axial chord downstream of the blade trailing edge,
the static pressure is specified and the density and velocity components
are extrapolated from the interior. At the solid surface of the blade
and hub, the no-slip condition is enforced, and temperature is specified
as per experimental data. The boundary conditions for turbulence
quantities on the walls are k = 0, and
c0 = 100 0u (16)
for a hydraulically smooth surface. An upper limit is imposed on the
value of co at the wall, as suggested by Menter (1992) and found
effective by Chima (1996),
800 v (17)(to,,_tx)_at= Re (Ay)2
The grid around the blade extends to mid-way between two
adjacent blades with periodic flow conditions in terms of cylindrical
velocity components set on a dummy grid line outside this boundary.
For a linear cascade (which is true for the experimental data), it is
possible to consider only half of the real span for computational
purposes with a symmetric boundary condition at mid-span.
Length Scale
The integral length scale describes the average eddy size
associated with the turbulence. For flow over a blade, its value affects
the passage vortex and thus the heat transfer considerably, as shown
later. Moreover, the value of the length scale to be used in
computation may differ from that measured experimentally depending
upon the definitions used. For the current experimental data (Giel et
al., 1999), an auto-correlation yielded a time scale which when
multiplied by the mean velocity yielded the integral length scale. If
is the computational length scale defined, as per Wilcox (1998), by
3Q(x'Y'Z't) = _ Jo k(x,y,z,t) dr
(18)
and fexp is the length scale measured experimentally (Giel et al., 1999)as
1 "fl_,(x,y,z,t,r) drQexp(X'Y'Z't) = 2 Jo k(x,y,z,t)
(19)
then [ = (3/8) [_. Here R, is the auto-correlation tensor for velocity,
and dr represents the infinitesimal displacement. We may mention that
according to Wilcox (1998), the relation between co, k and f is given
by Eq. (15).
EXPERIMENTAL DETAILS
Measurements were made in a linear cascade facility at the NASAGlenn Research Center (Giel et al., 1999). A turbine blade with 136 °
of turning, an axial chord of 127 mm and a span of 152.4 mm was
tested in a highly three-dimensional flow field resulting from thick
inlet boundary layers (cf. Table l). Data were obtained by a steady-
state technique using a heated, isothermal blade. Heat fluxes were
determined from a calibrated resistance layer in conjunction with a
surface temperature measured by calibrated liquid crystals. Data were
obtained for inlet Reynolds numbers of 0.5 and 1.0 x l06, for
isentropic exit Mach numbers of 1.0 and 1.3, and for inlet turbulence
intensities of 0.25% and 9.0%. More details are available in Giel et al.
(1999).
COMPUTATIONAL DETAILS
The computational span extended from the hub to mid-span of the
blade with a symmetric boundary condition at mid-span. In the axial
direction, the computational domain extended from the inlet plane
located one axial chord upstream of the blade leading edge to the exit
plane located 80% of the axial chord downstream of the blade trailing
edge. Around the blade, the grid extends to mid-way between two
adjacent blades with periodic boundary conditions. Figure 1 shows a
spanwise section of the multi-block viscous grid around the blade. The
viscous grid is obtained from an inviscid grid by clustering the grid
near all the solid walls (blade and hub here). The clustering is done
in such a way as to ensure that in the viscous grid, the distance of any
cell center adjacent to a solid wall, measured in wall units (y÷), is less
than half for the cases studied here, following Boyle and Giel (1992).
The average value for this distance was 0.26. The inviscid grid was
generated using the commercial code GridPro/az3000 (Program
Development Corporation, 1997). For computational accuracy the ratio
of two adjacent grid sizes in any direction was kept within 0.8-1.25.
NASA/CRy2001-210810 4
As canbeobserved from Fig. 1, the grid quality is very good
especially near the blade surface.
Initially, the grid consists of 28 blocks but before the solver is
used, it can be merged into just 5 blocks using the Method of Weakest
Descent (Rigby et aI., 1997). The final viscous grid consists of 366080
cells, formed by clustering near the blade and hub from an inviscid
grid with 55296 cells. The inviscid grid has 112 cells around the blade
(for the O-grid around the blade), 28 cells in the blade-to-blade
direction from the blade to the periodic boundary in-between the two
blades, and 16 in the spanwise direction. After clustering, the number
of cells in the spanwise direction increases to 52 and in the blade-to-
blade direction to 60. Three more grids were generated for a grid-
independence study. One inviscid grid had 1.5 times the number of
cells in each direction as compared to the basic grid described above.
Another inviscid grid had 32 cells in the spanwise direction while in
the other two directions, the number of cells were the same as in the
basic grid. For the third grid, the basic inviscid grid was clustered
near the blade and hub with a grid spacing half of that for the basic
viscous grid. All these variations of the basic grid yielded nearly the
same values for the heat transfer coefficient on the entire blade surface
as the basic grid; any variations were within ±2%. The results
presented here correspond to the basic grid shown in Fig. I.
Computations were run on the I6-processor C90 supercomputer
at NASA Ames Research Center. The code requires about 40 Mw of
storage with all blocks in memory, and takes about 15 s per iteration
for two levels of multi-grid. A case requires about 1200 iterations to
converge. Both the SST and k-to (Wilcox, 1998) models take almost
the same computational time and display similar numerical stability
characteristics. The Wilcox's 1988 k-to model takes somewhat less
time since the constants are simpler than those for the 1998 k-to model.
RESULTS and DISCUSSION
All the eight experimental cases for the GRC rotor were analyzed
for comparison. The values of various parameters for these cases are
given in Table 1. Figure 2 shows the static pressure distribution at
various spanwise locations on the blade surface for Case 3. While the
comparison between the computational and experimental values is very
good, we may note that the pressure distribution on the pressure
surface is little affected but on the suction surface is strongly affected
by the spanwise location. Moreover, the largest adverse pressure
gradient regions on the suction surface occur at mid-span, while along
the 10% spanwise location, there is almost no adverse pressure gradient
region on the suction surface. This may have some effect on the
prediction of heat transfer coefficient distribution on the suction surface
of the blade by the various turbulence models.
Before discussing the comparison between computed and
experimental data for heat transfer, however, let us first resolve the
issues about turbulence length scale and use of S2, _-" or Sf2 in Eq. (4)
for the production term. Figure 3 shows the effect of turbulence length
scale on the Stanton number distribution on the blade surface using the
k-to model of Wilcox (1998) for Re_. = 1.0 x 106, Mex = 0.98 and Tu_,
= 0.09. The entire blade surface is shown in Fig. 3 with the
streamwise distance s measured positive on the suction side starting at
the leading edge, and negative on the pressure side of the blade. In
this and later figures, both the streamwise distance s and the spanwise
distance z from the hub are normalized by the blade span. While
computations are done only over half the span (0 -< --,./span -<0.5), the
results are shown for the entire span, and depict clearly that the
symmetry boundary condition at mid-span is satisfied. The Stantonnumber, St, in this and later figures is calculated, as for the
experimental data, by
St = - k(c3T/On)w (20)
p,.V,oc/L - r,w)
where the local adiabatic wall temperature, Ta,, is
Taw 1 r-/-+
r,,_ l +0.50r l)_r:,(21)
M,, is the local isentropic Mach number, and the recovery factor r =
P//-_ (for turbulent flow) was used. According to Giel et al. (1999),
the overall uncertainty in the experimental data for St was determined
to be less than 13% in regions where St < 10 -_, and less than 8% where
St > 2 x 10 2. The computed results are shown in Fig. 3 for three
values of the turbulence length scale at inlet, namely _ = 0.03, 0.09
and 0.23. While the effect of _ on the pressure side of the blade is
minor, that on the suction side of the blade is considerable, and stems
from its influence on the passage vortex flow, which is essentially a
three-dimensional flow phenomenon. As the value of _ decreases, the
extent of the passage vortex flow increases. This raises the heattransfer coefficient on the entire suction side of the blade, and the
effect is quite pronounced in the passage vortex region near the hub.
For Wilcox (1988) k-o_ model, results are very similar to those in Fig.
3. Differences between the Stanton number contours resulting from the
two k-to models are too small to be displayed here.
The effect of using S" or fl'- or S_ in the production term for Eqs.
(1) and (2) is shown in Fig. 4 in terms of the Stanton number
distribution on the blade surface. Results are shown for the SST model
for Case 1 (Re_, = 106, M,x = 0.98, Tuin = 0.09 and _ = 0.08625, which
corresponds to experimental Q = 0.23). Use of S 2 leads to very high
heat transfer coefficient at the leading edge of the blade, as is well
known and is shown clearly in Fig. 4(a). This also increases the heattransfer coefficient over the entire blade surface. Use of either 122 or
Si2 leads to small differences in the Stanton number distributions on
the blade surface. Comparison of the Stanton number distribution with
the experimental data at three spanwise locations, shown in Fig. 5,
reveals that use of S_ in the production term produces results in closer
agreement with the experimental data than that of _". Thus, S_ was
used in place of S2 in Eq. (4) for the results that follow.
Figures 4(c) and 6 show the differences between the SST model
and Wilcox (1998) k-o> model in terms of Stanton number distribution
on the blade surface for Case 1. While the differences on the pressure
surface are minor, those on the leading edge and on the suction surface
are appreciable. The SST model yields a bigger passage vortex region
near the hub as compared to the k-to model. This results in higher
Stanton number values near the hub but slightly lower values near the
mid-span on the suction surface of the blade for the SST model. This
is clearly evident from Figs. 5(a) and 7 which provide a comparison
with the experimental data at three spanwise locations (10%, 25% and
50%) on the blade for this case. While the computed Stanton number
values based on either of the two models compare very well with the
NASA/CR--2001-210810 5
experimentaldata on the pressure surface, there are discrepancies on
the suction surface. The SST model yields a better comparison with
the experimental data than the k-o) model at 10% and 25% spanwise
locations, while the k-o) model yields a somewhat better comparison
at mid-span. This appears to be a direct consequence of two
phenomena: l) a better resolution of the passage vortex region by the
SST model, and 2) a better prediction by the k-o) model in large
adverse pressure gradient regions. We may recall (cf. Fig. 2) that the
largest adverse pressure gradient region lies near mid-span on the
suction side of this rotor.
Figure 8 shows the differences between the SST model and
Wilcox (1998) k-o) model in terms of Stanton number distribution on
the blade surface for Case 3. Here, the SST model yields slightly
higher Stanton number values on the pressure surface of the blade, but
in contrast to Case 1 in Figs. 4(c) and 6, the k-t0 model yields a bigger
passage vortex region near the hub as compared to the SST model.
This results in higher Stanton number values near the hub but slightly
lower values near the mid-span on the suction surface using the k-o)
model. This is evident from Fig. 9 which provides a comparison with
the experimental data at three spanwise locations (10%, 25% and 50%)
on the blade for this case. While the computed Stanton number values
based on the SST model compare very well with the experimental data
on the pressure surface, those using the k-o) model do not compare that
well. The SST model yields a better comparison with the experimental
data than the k-o) model at mid-span, while the k-to model yields a
somewhat better comparison at 10% and 25% spanwise locations.
These trends (in terms of heat transfer prediction by the SST and k-o)
models) are in direct contrast to the observations made above for Case
1. Since the SST model provides a better overall comparison with the
experimental data, rest of the cases in Table 1 were run using the SST
model only.
Figure 10 provides a comparison with the experimental data for
Cases 2 and 4 at three spanwise locations using the SST model. While
the Stanton number values compare well on the pressure side of the
blade and at the leading edge, there are quantitative differences
between the computed and experimental data on the suction side. The
qualitative nature of the predictions compares well with the
experimental data except that interaction of the shock wave with the
boundary layer downstream of the throat (at s/span -- 1.25) is not
predicted by the SST model.
Figure 11 compares the experimental data for Cases 5 and 6 at
three spanwise locations with the results computed using the SST
model. Again, the comparison is very good on the pressure side of the
blade and at the leading edge, but it is not so good on the suction side
of the blade. For the transonic Case 6, the SST model does not predict
the shock wave downstream of the throat, as for Cases 2 and 4 in Fig.
10.
Figure 12 compares the experimental data for Cases 7 and 8 at
three spanwise locations with the results computed using the SST
model. Here, the comparison is not good over the entire blade surface,
including the leading edge of the blade. Both these cases are for very
low turbulence level (Tai, = 0..0025),), whic_ is not practicaV_ real _
turbines. Thus, a poor comparison for these cases should not be of
much concern. We may also note that the Reynolds number is half of
that for cases 3 and 4 discussed earlier for the low turbulence level.
CONCLUSIONS
Two versions of the two-equation k-o) model and a shear stress
transport (SST) model are used in a three-dimensional, multi-block,
Navier-Stokes code to compare the detailed heat transfer measurements
(Giel et al., 1999) on a transonic turbine blade. It is found that the SST
model resolves the passage vortex better on the suction side of the
blade, thus yielding a better comparison with the experimental data
than either of the k-o) models. However, in large adverse pressure
gradient regions, such as in the mid-span region on the suction side of
this blade, the prediction of heat transfer coefficient by the SST model
may be poorer than that by the k-o) model. Moreover, the prediction
by any of the turbulence models tested here is still deficient on the
suction side of the blade.
Use of the SST model does require the computation of distance
from a wall, which for a multi-block grid, such as in the present case,
can be complicated. However, a relatively easy fix for this problem
was devised. Also, both the k-o) and SST models were found to have
similar numerical stability and convergence characteristics. Some
issues related to the application of two-equation turbulence models
have also been resolved. For example, upon comparison with the
experimental data, it is found that for aerodynamic applications use of
S_ is better than using either S" or f2"- for the computation of
production term in the turbulence equations. Also, the turbulence
length scale influences the passage vortex on the suction side of the
blade, and thus the relation between the computational and
experimental values of the length scale becomes important. For the
present study, the two values of the length scale are related via a factor
of 3/8.
REFERENCES
Boyle, R.J. and Giel, P., 1992, Three-Dimensional Navier Stokes
Heat Transfer Predictions for Turbine Blade Rows, AIAA Paper 92-
3068.
Chima, R.V., 1996, A k-o) Turbulence Model for Quasi-Three-
Dimensional Turbomachinery Flows, AIAA Paper 96-0248.
Giel, P.W., Van Fossen, G.J., Boyle, R.J., Thurman, D.R. &
Civinskas, K.C., 1999, Blade Heat Transfer Measurements and
Predictions in a Transonic Turbine Cascade, ASME Paper 99-GT-125.
Jameson, A., Schmidt, W. and Turkel, E., 1981, Numerical
Solutions of the Euler Equations by Finite Volume Methods Using
Runge-Kutta Time-Stepping Schemes, AIAA Paper 81-1259.
Jones, W.P. and Launder, B.E., 1973, The Calculation of
Low-Reynolds-Number Phenomena with a Two Equation Model of
Turbulence, Int. J. Heat Mass Transfer, vol. 16, pp. 1119-1130.
Kato, M. and Launder, B.E., 1993, The Modelling of Turbulent
Flow Around Stationary and Vibrating Square Cylinders, Proc. 9th
Syrup. Turbulent Shear Flows, Kyoto, Japan, 10-4-1.
Menter, F.R., 1992, Improved Two-Equation k-o) Turbulence
Models for Aerodynamic Flows, NASA TM 103975.
Menter, F.R., 1994, Two-Equation Eddy-Viscosity Turbulence
Models for Engineering Applications, AIAA J., vol. 32, pp. 1598-1605.
Menter, F.R., 1996, A Comparison of Some Recent Eddy-
Viscosity Turbulence Models, J. Fluids Eng., vol. 118, pp. 514-519.
NASA/CR--2001-210810 6
Moore,J.G.andMoore,J.,1999,RealizabilityinTurbulenceModelingforTurbomachineryCFD,ASMEPaper99-GT-24.
Moss,R.W.andOldfield,M.L.G.,1992,Measurementsof theEffectof Free-StreamTurbulenceLengthScaleonHeatTransfer,ASMEPaper92-GT-244.
ProgramDevelopmentCorporation,1997,"GridProa_/az3000-User'sGuideandReferenceManual,"WhitePlains,NY.
Rigby,D.L.,Steinthorsson,E.andCoirier,W.J.,1997,Automatic
Block Merging Using the Method of Weakest Descent, AIAA Paper
97-0197.
Schlichting, H., 1979, Boundary Layer Theory, McGraw-Hill,
New York, 7 _ edition, pp. 312-313.
Steinthorsson, E., Ameri, A.A. and Rigby, D.L., 1997,
TRAF3D.MB - A Multi-Block Flow Solver for Turbomachinery Flows,
AIAA Paper 97-0996.
Van Fossen, G.J., Simoneau, R.J. and Ching, C.Y., 1995,
Influence of Turbulence Parameters, Reynolds Number, and Body
Shape on Stagnation-Region Heat Transfer, J. Heat Transfer, vol. 117,
pp. 597-603.
Wilcox, D.C., 1988, Reassessment of the Scale-Determining
Equation for Advanced Turbulence Models, AIAA J., vol. 26, pp. 1299-
1310.
Wilcox, D.C., 1998, Turbulence Modeling for CFD, DCW
Industries, 2nd Edn.
Case
1
2
3
4
5
6
7
8
Table 1 Parameter Values
Blade axial chord = 127 mm (5.0 in)
Blade span = 152.4 mm (6.0 in)
Inlet Mach number = 0.38
TJ'Fo_ . = 1.085 for blade; = 1.0 for hub
Rein
1.0 × 106
1.0 x 106
1.0 x 106
1.0 × 106
0.5 × 106
0.5 x 106
0.5 × 106
0.5 × 106
Mcx
0.98
1.32
0.98
1.32
0.98
1.32
0.98
1.32
InletTu_. Expfl. Q
0.09 0.23 20.3 mm
0.09 0.23 20.3 mm
30.5 mm0.0025 0.01
0.0025 0.0i 30.5 mm
0.09 0.23 20.3 mm
0.09 0.23 20.3 mm
0.0025 0.01 30.5 mm
0.0025 0.01 30.5 mm
NASA/CR--2001-210810 7
iJ._
Figure 1.-Spanwise section of the multi-block viscous gridaround the rotor.
1.1
0.9
0_- 0.7
0.5
0,30
. I I I I I I l I l
_-_':'..-_"-----,-......_ "_'_. \\ -I
•', S%exp_ "" ....._1a 2_5%o expt T-I
-- 50% calc '_1....... 25% calc I J I I I 7
..... 10% calc 0.6 0.8 1.0
..... 5% calc
.......... 2.5% calc x/Cxi
Figure 2.- Static pressure distribution on the blade surface
at various spanwise locations for Case 3,|
NASA/CRy2001-210810 8
1.0
o.s ,
0.0 ....
(a)
1.0
==o.s
0•0
(b)
St x 1000
, ,l,_,_l+qP,l,_q_l_qn,l_JHllii_l,,_tlP_blq ,, ,IH HI,, HI,,,,I,tn,IH
0.8 -0.4 0.0 0.4 0.8 1.2 1.6
Pressure s/span Suction
St x 1000
_ll_bb111tl')_''_Ib'b'l'_'bl_lHl'''ll_H _It'I'l'J''_H Pqlq 'q_l_' ''I''_Pl '
-0•8 -0.4 0.0 0.4 0.8 1•2 1.6
Pressure s/span Suction
1.0 -I _ St x 1000
I I I I I I I I I I I I I I I
(c) -0.8 -0.4 0•0 0•4 0.8 1•2 1.6Pressure s/span Suction
Figure 3.- Effect of turbulence length scale on Stantonnumber distribution on the blade using the k- oomodel(1998) for Rein = 1.0x106; Mex = 0.98; Tuin = 0.09•(a) Length scale = 0.03. (b) Length scale = 0.09.(c) Length scale = 0.23.
I•0 l St x 1000
(a) -0.8 -0.4 0.0 0.4 0.8 1.2 1.6Pressure s/span Suction
I05-
(b) -0.8 -0.4 0.0 0.4 0•8 1.2 1.6Pressure s/span Suction
1.0 ] St x 1000
(c) -0.8 -0.4 0.0 0.4 0.8 1.2 1.6Pressure s/span Suction
Figure 4.- Stanton number distribution on the blade usingthe SST model for Case 1. (a) with S2; (b) with V2; (c) with
SV for production term In Eq. (4).
NASA/CR--2001-210810 9
5
X
4
50% expt
7 -I- v 25% expt
- [] 10% expt_ 50% calc
6 ...... 25% calc
........ 10% calc
w
- (a)I I
-1 0 1s/span
I f "1 I I I I i i -
,, • ; °, , __
_v_ I
,,\/ \
6
4
3
-I_ M 50% expl7- v 25%expt I I I I I I I I I I _
[] 10% expt od"'o"_[o...50% calc o
" ..... 25% ca c q_ o _z. _-,¢%_ --..--_.D 10% calc _ ,"'. __.
Ew _ _ O: \ ---- M I MI_ t ; . j- I _'#,_-m _ "_"....
- _ _,,_# _ '.... --I- lY
-1 0 1 2s/span
Figure 5.- Comparison with experimental data at three spanwiselocations using the SST model for Case 1. (a) with SD, (b) with
_2 for production term in Eq. (4).
NASA/CR--2001-210810 10
1.0 _ Stx 1000
-0.8 -0.4 0.0 0.4 0.8 1,2 1.6Pressure s/span Suction
Figure 6.- Stanton number distribution on the blade surface for Case 1 usingk- (o model (1998).
x 50% expt
7 -I- v 25% expt
- [] 10% expt_ 50% calc
6 ...... 25% calc
5
X
I I I I I I I I I I _
go% -[] _ -2_
; ....... 10% talc _ [] _[] _
: I_1_. _ _- _- _".'__"I,4 - i\\_, ,_ '1:::'._,,, #,.." ,,, "-......_
- 0 1 2s/span
Figure 7.- Comparison with experimental data at three spanwiselocations using the k- ¢omodel (1998) for Case 1.
NASA/CR--2001-210810 l I
1.0
¢.-
_0.5"N
0.0
(a)
Stx 1000
/_/_/tl _ __
-- T,,I,,,,I,,,,I,,,,I,,,,I,,,,I,,,,I,,,,I,,,
-0.8 -0.4 0.0 0.4 0.8 1.2 1.6
Pressure s/span Suction
St x 1000
10 iO0 r,,,''"l""l""l''"l''"l "l""l""l' I t t t t I _
-0.8 -0.4 0.0 0.4 0.8 1.2 1.6
(b) Pressure s/span Suction
Figure 8.- Stanton number distribution on the blade surface for Case 3 using(a) SST model, and (b) k-¢o model (1998).
NASAJCR_2001-210810 12
5
X
7 i _
- E 50% expt
- v 25% expt
- [] 10% expt
50% calc
..... 25% calcm
....... 10% calc
I
I| I I I I i1 I
-1
I I I I i I I I I
vv v[] .--_,_ v vvo/
V t_r
%. , , ,,
•E_:]=..g_M_. / VVv v-,- ..... .L, I mk-_ " /'*_ v a,, "_. i
I I I I I I I I I
0 1 2
s/span
7!111111 llllllllll
i 50% expt
v 25% expt n°q:t_
[] 10% expt _ -
5 - 50% calc _"/"_,v "_ v --
..... 25% calc _ T_-_" vvv_
.......'°_"'_ _,. _ ,__:'x,_H"i-
t:.....,..'_----L_ -_;_'_-, ! -
-1 0 1 2
s/span
Figure 9.- Comparison with experimental data at three spanwise
locations using (a) SST model, and (b) k- oJmodel (1998) for Case 3.
NASA/CR--2001-210810 13
¢-
x
7
6
5
4
3
2
50% expt I
10% expt cFPq_50% calc
...... 25% calc v o10% calc /_,_.D_ o /':
m
, i i_i _ i ",.u
- _ _'::" _ __,__'._ _
_ _\ .,_ v ;! "_-
--
-I I I i l i i I I I I I I I "1_ .-,l
-I 0 1
s/span
5
o
x
- , " 50% expt. , v 25% expt o°C_
O 13[] 10% expt o
- I
50% calc oo 0 _ [] COL
....... 25_calc o ,._ o
I- I ....... 10% calc o [ "v t( _iV : _ V u
' _ _ -- co-_o, _, o v\ _ :v _ v
.__ , ,_vCL,L - 7".,_. v,,,.
-1 0 1
s/span
_ T r--_--_--
2
Figure 10.- Comparison with experimental data at three spanwiselocations using the SST model. (a) Case 2; (b) Case 4.
NASA/CR--2001-210810 14
7 -T
- o 10% expt!
- -- 50% calc I
6 - ..... 25% calc I ¢ \"_"........ 10% calc .0 '_.d .,,
4 -- i _
3 -- /
: , \,/_(a)j I i J [ J I i I I =l I t I [
2-1 0 1
s/span
M 50% expt
, %exptI''' '-Q....,
D
I-
_ 5
X
_ 4
= 50% expt
-I v 25% expt
- [] 10% expt_ 50% calc
- - .... 25% calc
....... 10% calc-I
- '\
- 'X
-- . %,%%%
;l(b)l I _ _ _ _ 1-1 0
' _ i i i i I i i i i Z111
Ii_., °i ._ n,._.'=k i .,..[]n
_',, i _v'-'_-
- " ,,_ i_,,,, I, _ ,--_I_
1 2s/span
Figure 11.- Comparison with experimental data at three spanwiselocations using the SST model. (a) Case 5; (b) Case 6,
NASA/CR--2001-210810 15
8o
x
8 7m
m
n
I
i 50%expt I t t t I I I I vl Iv 25% expt v_
_IDO ,Eo 10% expt . _._
25%50%ca,Cca,c --_ g_.l_ r_VC#J "-'o,
....... 10% calc -i o!_
oi ;"_ I_1 ]Dl[,_r_. . I "
., 2111¢_!v I " i
' j
}a)l t
-1 0 1
s/span
Im
i
2
8 -F _ 50%expt J I I t I- v 25% expt
- o 10% expt
---: 50% calc t6-- 25% calc n_vv_
-I ....... 10%calc c_ oi JL \o _v t
- I_,', 5 _,#,' ',_,w
" :'- J _'A_
-i(_)iJ I i _l i i I I I , I I r J i
-1 0 1
s/span
Figure 12.- Comparison with experimental data at three spanwiselocations using the SST model. (a) Case 7; (b) Case 8.
4
2
0
I I I I
13
n_13 -cl::_q_ m
I --
I
2
NASA/CRy2001-210810 16
REPORT DOCUMENTATION PAGE FormApprovedOMB No. 0704-0188
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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED
4. TITLE AND SUBTITLE
April 2001
Two-Equation Turbulence Models for Prediction of Heat Transfer
on a Transonic Turbine Blade
6. AUTHOR(S)
Vijay K. Garg and All A. Ameri
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
AYT Corporation
2001 Aerospace Parkway
Brook Park, Ohio 44142
9. SPONSORINGu_ONITORING AGENCY NAME(S) AND ADDRESS(ES)
National Aeronautics and Space Administration
Washington, DC 20546- 0001
Final Contractor Report
5. FUNDING NUMBERS
WU-714-03 -50--00
NAS3-00180
8. PERFORMING ORGANIZATION
REPORT NUMBER
E-12717
10. SPONSORING/MONITORING
AGENCY REPORT NUMBER
NASA CR--2001-210810
2001-GT-0165
11. SUPPLEMENTARY NOTES
Prepared for the 2001 Turbo Expo sponsored by the American Society of Mechanical Engineers, New Orleans, Louisiana,
June 4-7, 2001. Project Manager, R.E. Gaugler, Turbomachinery and Propulsion Systems Division, NASA Glenn
Research Center, organization code 5820, 216--433-5882.
12a, DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE
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Subject Categories: 34 and 07 Distribution: Nonstandard
Available electronically at htm://eltrs, zrc.nasa. _ov/GLTRS
This _ublication is available from the NASA Center for AeroSpace Information, 301--621-0390.
13. ABSTRACT (Maximum 200 words)
Two versions of the two-equation k-o) model and a shear stress transport (SST) model are used in a three-dimensional,
multi-block, Navier-Stokes code to compare the detailed heat transfer measurements on a transonic turbine blade. It is
found that the SST model resolves the passage vortex better on the suction side of the blade, thus yielding a better
comparison with the experimental data than either of the k-o) models. However, the comparison is still deficient on the
suction side of the blade. Use of the SST model does require the computation of distance from a wall, which for a multi-
block grid, such as in the present case, can be complicated. However, a relatively easy fix for this problem was devised.
Also addressed are issues such as (1) computation of the production term in the turbulence equations for aerodynamic
applications, and (2) the relation between the computational and experimental values for the turbulence length scale, and
its influence on the passage vortex on the suction side of the turbine blade.
14. SUBJECT TERMS
Heat transfer; Turbine; Turbulence modeling
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